College algebra, 9th edition

Review “Study for Quizzes and Tests” Feature Description Benefit Page Chapter Reviews at the end of each chapter contain… “Things to Know” A detailed list of important theorems, formulas, and definitions from the chapter Review these and you’ll know the most important material in the chapter! 494–495 “You Should Be Able to…” Contains a complete list of objectives by section, examples that illustrate the objective, and practice exercises that test your understanding of the objective Do the recommended exercises and you’ll have mastery over the key material If you get something wrong, review the suggested page numbers and try again 495–496 Review Exercises These provide comprehensive review and practice of key skills, matched to the Learning Objectives for each section Practice makes perfect These problems combine exercises from all sections, giving you a comprehensive review in one place 496–499 CHAPTER TEST About 15–20 problems that can be taken as a Chapter Test Be sure to take the Chapter Test under test conditions—no notes! Be prepared Take the sample practice test under test conditions.This will get you ready for your instructor’s test If you get a problem wrong, watch the Chapter Test Prep video 500 CUMULATIVE REVIEW These problem sets appear at the end of each chapter, beginning with Chapter They combine problems from previous chapters, providing an ongoing cumulative review These are really important They will ensure that you are not forgetting anything as you go These will go a long way toward keeping you constantly primed for the final exam 500–501 CHAPTER PROJECTS The Chapter Project applies what you’ve learned in the chapter Additional projects are available on the Instructor’s Resource Center (IRC) The Project gives you an opportunity to apply what you’ve learned in the chapter to solve a problem related to the opening article If your instructor allows, these make excellent opportunities to work in a group, which is often the best way of learning math 501–502 In selected chapters, a web-based project is given The projects allow the opportunity for students to collaborate and use mathematics to deal with issues that come up in their lives 501–502 NEW! Internet-based Projects This page intentionally left blank To the Student As you begin, you may feel anxious about the number of theorems, definitions, procedures, and equations You may wonder if you can learn it all in time Don’t worry, your concerns are normal This textbook was written with you in mind If you attend class, work hard, and read and study this book, you will build the knowledge and skills you need to be successful Here’s how you can use the book to your benefit Read Carefully When you get busy, it’s easy to skip reading and go right to the problems Don’t the book has a large number of examples and clear explanations to help you break down the mathematics into easy-to-understand steps Reading will provide you with a clearer understanding, beyond simple memorization Read before class (not after) so you can ask questions about anything you didn’t understand You’ll be amazed at how much more you’ll get out of class if you this Use the Features I use many different methods in the classroom to communicate Those methods, when incorporated into the book, are called “features.” The features serve many purposes, from providing timely review of material you learned before (just when you need it), to providing organized review sessions to help you prepare for quizzes and tests Take advantage of the features and you will master the material To make this easier, I’ve provided a brief guide to getting the most from this book Refer to the “Prepare for Class,” “Practice,” and “Review” pages on the inside front cover of this book Spend fifteen minutes reviewing the guide and familiarizing yourself with the features by flipping to the page numbers provided Then, as you read, use them This is the best way to make the most of your textbook Please not hesitate to contact me, through Pearson Education, with any questions, suggestions, or comments that would improve this text I look forward to hearing from you, and good luck with all of your studies Best Wishes! Michael Sullivan Step-by-step solutions on video for all chapter test exercises from the text English Subtitles Available CHAPTER TEST PREP VIDEOS AR E ACCE SSIBLE THROUGH THE FO LLOWING: Dedicated to the Memory of Mary This page intentionally left blank COLLEGE ALGEBRA NINTH EDITION This page intentionally left blank COLLEGE ALGEBRA NINTH EDITION Michael Sullivan Chicago State University Prentice Hall Index Difference(s), 7, 11–12 common, 647 of complex numbers, 105 of logarithms, 453 of two cubes, 44, 50–51 of two functions, 208–10 of two matrices, 582–83 of two squares, 43, 50–51 Difference quotient, 205, 234, 436 Diophantus, 660 Directrix, of parabola, 505 Direct variation, 189, 191 Dirichlet, Lejeune, 199 Discontinuity, 239 Discriminant, 97 negative, 109 Disjoint sets, Distance mean, 523, 539 on real number line, 19–20 Distance formula, 150–53 proof of, 151–52 using, 152 Distributive Property of matrix multiplication, 588 of real numbers, 10 Divergent geometric series, 656–58 Dividend, 45, 375 Division, of complex numbers in standard form, 107 in order of operations, of polynomials, 44–47 algorithm for, 375 synthetic, 58–62 properties of, 12 of rational expressions, 63–64 of two integers, 45 Divisor, 45, 375 Domain, 201, 206–8 of absolute value function, 238 of composite function, 402–5 of constant function, 237 of cube function, 237 of cube root function, 237 defined by an equation, 207 of difference function, 208 of greatest integer function, 238 of identity function, 237 of inverse function, 412 of logarithmic function, 439–40 of logistic models, 481 of one-to-one function, 409 of product function, 208 of quotient function, 209 of rational function, 343–46 of reciprocal function, 238 of square function, 237 of square root function, 237 of sum function, 208 unspecified, 210 of variable, 21 Domain-restricted function, 416–17 Doppler, Christian, 366 Doppler effect, 366 Dot mode, 239 Double root (root of multiplicity 2), 93 Dry adiabatic lapse rate, 652 e, 428–30, 436 defined, 429 Earthquakes, magnitude of, 450 Eccentricity of ellipse, 524 of hyperbola, 536 Effective rates of return, 469–70 Egyptians, ancient, 100, 660 Elements (Euclid), 660 Elements of sets, 2, 680–82 Elimination, Gauss-Jordan, 563 Elimination method, 540, 545–46, 547–48 systems of nonlinear equations solved using, 606–10 Ellipse, 504, 514–24 with center at (h, k), 518–19 with center at the origin, 514–18 major axis along x-axis, 515–16 major axis along y-axis, 517 with center not at origin, 519–20 center of, 514 defined, 514 eccentricity of, 524 foci of, 514 graphing of, 516–19 major axis of, 514 length of, 514 minor axis of, 514 solving applied problems involving, 520–21 vertices of, 514 Ellipsis, Elliptical orbits, 503 Empty (null) sets, 2, 680 End behavior, 330–32 of rational function, 346 Endpoints of interval, 120 Entries of matrix, 556, 581 diagonal, 588 Equality of complex numbers, 105 properties of, of sets, 2, 680 Equally likely outcomes, 696–97 I5 I6 Index Equal sign, Equation(s) demand, 300 depressed, 379 domain of a function defined by, 207 equivalent, 83, A3 even and odd functions identified from, 224 exponential, 430–32, 444, 461–62 quadratic in form, 462 as function, 203 graphing utility to graph, A3–A4 historical feature on, 89 intercepts from, 160 inverse function defined by, 414–17 involving absolute value, 130 quadratic in form, 114–16 solving, 114–16 satisfying the, 82, 157 sides of, 82, 157 solution set of, 82 solving, 82 with graphing calculator, A6–A7 in two variables, graphs of, 157–67 intercepts from, 159 by plotting points, 157–59 symmetry test using, 160–62 x ϭ y2, 163 y ϭ Ϭ x, 164 y ϭ x3, 164 Equilateral hyperbola, 536 Equilateral triangle, 155 Equilibrium price, 277–78 Equilibrium quantity, 277–78 Equivalent equations, 83, A3 Equivalent inequalities, 121, 123, 126 Equivalent systems of equations, 545 Error triangle, 156 Euclid, 100, 660, 673 Euler, Leonhard, 199, 701 Even functions determining from graph, 223 identifying from equation, 224 Evenness ratio, 449 Events, 696 complement of, 699–700 mutually exclusive, 698–99 probabilities of union of two, 698–99 Explicit form of function, 206 Exponent(s), 22 Laws of, 21–23, 422, 431 logarithms related to, 438 Exponential equations, 430–32 defined, 430 solving, 430–32, 444, 461–62 equations quadratic in form, 462 using graphing utility, 462–63 Exponential expressions, changing between logarithmic expressions and, 438 Exponential functions, 421–37 defined, 423 e, 428–30, 436 evaluating, 421–25 fitting to data, 487–88 graph of, 425–28 using transformations, 428, 429–30 identifying, 423–25 power function vs., 423 properties of, 426, 428, 432 ratio of consecutive outputs of, 423–24 Exponential growth and decay, 422, 476–86 law of decay, 478–79 logistic models, 481–83 defined, 481 domain and range of, 481 graph of, 481 properties of, 481 uninhibited growth, 476–78 Exponential law, 476 Extended Principle of Mathematical Induction, 667 Extraneous solutions, 113 Extreme values of functions, 227 Extreme Value Theorem, 227 Factored completely, 50 Factorial symbol, 639–40 Factoring defined, 49 of expression containing rational exponents, 77 over the integers, 50 polynomials, 49–58 Ax2 + Bx + C, 54–55 difference of two squares and the sum and the difference of two cubes, 50–51 by grouping, 53–54 perfect squares, 51–52 x2 + Bx + C, 52–53 quadratic equations, 93–95, 116–17 Factors, 7, 49 linear, 598–602 nonrepeated, 598–99 repeated, 600–601 quadratic, 381, 602–3 synthetic division to verify, 61 Factor Theorem, 375–77 Family of lines, 181 of parabolas, 255 Feasible point, 622, 623–24 Fermat, Pierre de, 149, 437, 700 Ferrari, Lodovico, 384 Index Ferris, George W., 187 Fertility rate, 636 Fibonacci, 660 Fibonacci numbers, 641 Fibonacci sequences, 641, 645 Financial models, 466–75 compound interest, 466–72 doubling time for investment, 471 effective rates of return, 469–70 future value of a lump sum of money, 466–69 present value of a lump sum of money, 468, 470–71 tripling time for investment, 472 Finite sets, 680 Fixed costs, 180 Focus/foci of ellipse, 514 of hyperbola, 524 of parabola, 505 FOIL method, 43 Formulas, geometry, 31–32 Foucault, Jean Bernard Leon, 119 Fractions continued, 72 least common multiple to add, 14 partial, 598 Frobenius, Georg, 594 Function(s), 199–270 absolute value, 236, 238 argument of, 204 average cost, 217 average rate of change of, 228–30 finding, 228–30 secant line and, 229–30 building and analyzing, 257–63 on calculators, 205–6 constant, 224–25, 226, 236–37 continuous, 239, 382 cube, 204, 237 cube root, 235, 237 decreasing, 224–25, 226, 228 defined, 201 difference of two, 208 difference quotient of, 205 domain of, 201, 206–8 unspecified, 210 domain-restricted, 416–17 equation as, 203 even and odd determining from graph, 223 identifying from equation, 224 explicit form of, 206 graph of, 214–22, 244–57 combining procedures, 247, 252 determining odd and even functions from, 223 determining properties from, 224–25 identifying, 214–15 information from or about, 215–17 using compressions and stretches, 247–49, 251 using reflections about the x-axis or y-axis, 250–51 using vertical and horizontal shifts, 244–47, 251 greatest integer, 238–39 identity, 237 implicit form of, 206 important facts about, 206 increasing, 224–25, 226, 228 library of, 234–39 local maxima and local minima of, 225–26 nonlinear, 273 objective, 622–26 one-to-one, 408–11 piecewise-defined, 239–41 power, 321–24 graph of, 322–23 of odd degree, 323 properties of, 323 product of two, 208 quotient of two, 209 range of, 201 reciprocal, 238 relation as, 200 square, 237 square root, 234–35, 237 step, 239 sum of two, 208 value (image) of, 201, 203–6 zeros of, Bisection Method for approximating, 387 Function keys, Function notation, 210 Fundamental Theorem of Algebra, 388 Conjugate Pairs Theorem and, 389 proof of, 388 Future value, 466–69 Galois, Evariste, 384 Gauss, Karl Friedrich, 388, 540 Gauss-Jordan method, 563 General addition principle of counting, 682 General form of equation of circle, 184–85 linear equation in, 174–75 General term, 638 Generators of cone, 504 Geometric mean, 129 Geometric sequences, 653–56 common ratio of, 653 defined, 653 determining, 653–54 I7 I8 Index Geometric sequences, (Continued) formula for, 654–55 nth term of, 654–55 sum of, 655–56 Geometric series, 656–60 infinite, 656–57 Geometry essentials, 30–38 formulas, 31–32 Pythagorean Theorem and its converse, 30–31, 35 Geometry problems, algebra to solve, 153 George I of Greece, King, 149 Golden ratio, 645–46 conjugate, 646 Grade, 181 Graph(s)/graphing bounded, 618 of circles, 183–84 complete, 159 of ellipse, 516–19 of equations in two variables, 157–67 intercepts from, 159 by plotting points, 157–59 symmetry test using, 160–62 x ϭ y2, 163 y ϭ ÷ x, 164 y ϭ x3, 163 of exponential functions, 425–28 using transformations, 428, 429–30 of function, 214–22, 244–57 combining procedures, 247, 252 determining odd and even functions from, 223 determining properties from, 224–25 identifying, 214–15 information from or about, 215–17 in library of functions, 234–39 using compressions and stretches, 247–49, 251 using reflections about the x-axis or y-axis, 250–51 using vertical and horizontal shifts, 244–47, 251 of inequalities, 18–19, 614–18 linear inequalities, 615–16 steps for, 615 of inverse functions, 413–14 of lines given a point and the slope, 170 using intercepts, 174–75 to locate absolute maximum and absolute minimum of function, 226–27 of logarithmic functions, 440–43 base not 10 or e, 456 inverse, 441–43 of logistic models, 481–83 of parabola, 506 of piecewise-defined functions, 239–41 of polynomial functions, 322–37 analyzing, 332–36 end behavior of, 330–32 smooth and continuous, 321 turning points of, 329–30 using bounds on zeros, 382 using transformations, 324 using x-intercepts, 326–27 of polynomial inequalities, 368 of quadratic functions properties of, 292–96 steps for, 296 using its vertex, axis, and intercepts, 292–96 using transformations, 290–92 of rational functions, 353–67 analyzing, 353–63 constructing rational function from, 363–64 end behavior of, 346 using transformations, 344 of rational inequalities, 370 of sequences, 637–38 to solve systems of equations, 543 of systems of nonlinear inequalities, 617–18 of y = a b , 344 x Graphing calculator(s), caret key on, 24 composite functions on, 402 exponents evaluated on, 24 Graphing utility(ies), A1–A10 connected mode, 239 coordinates of point shown on, A2 dot mode, 239 eVALUEate feature, 376, A5 to find sum of arithmetic sequence, 649 to fit exponential function to data, 487–88 to fit logarithmic function to data, 488–89 to fit logistic function to data, 489–90 functions on, 228 geometric sequences using, 655, 656 to graph a circle, 185 to graph equations, A3–A4 to graph inequalities, A9 graph of polynomial function analyzed with, 335–36 INTERSECT feature, A6–A7 line of best fit from, 284–85 to locate intercepts and check for symmetry, A5–A6 logarithmic and exponential equations solved using, 462–63 matrix operations on, 583 MAXIMUM and MINIMUM features, 228 REF command, 567 REGression options, 487 RREF command, 567, A10 to solve equations, A6–A7 to solve systems of linear equations, A9–A10 square screens, A8 TABLE feature, 383, 638 Index tables on, A4 TRACE feature, 638 turning points in, 329 viewing rectangle, A1–A3 setting, A1 ZERO (or ROOT) feature, 303, A5, A6 ZOOM-STANDARD feature, A3n ZSquare function on, A8n Greatest integer function, 238–39 Greeks, ancient, 15 Grouping, factoring by, 53–54 Growth, uninhibited, 476–78 Growth factor, 423 Hale-Bopp comet, orbit of, 503, 539 Half-life, 478 Half-open/half-closed intervals, 120 Half-planes, 615 Harmonic mean, 129 Harriot, Thomas, 100 Heron of Alexandria, 660 Hindus, ancient, 100 Horizontal asymptote, 345–46 Horizontal compression or stretches, 249 Horizontal lines, 171–72 Horizontal-line test, 410–11 Horizontal shifts, 244–47, 251 HP 48G, A8n Huygens, Christiaan, 700 Hyperbolas, 503, 504, 524–36 asymptotes of, 529–31 branches of, 524 with center at (h, k), 531–32 with center at the origin, 524–29 transverse axis along x-axis, 526–27, 531 transverse axis along y-axis, 527–28, 531 with center not at the origin, 531–32 center of, 524 conjugate, 536 conjugate axis of, 524 defined, 524 eccentricity of, 536 equilateral, 536 foci of, 524 graphing equation of, 526–27 solving applied problems involving, 532–33 transverse axis of, 524 vertices of, 524 Hyperbolic cosine function, 437 Hyperbolic sine function, 437 Hyperboloid, 536 Hypotenuse, 30 i, 104–5 powers of, 108–9 Ibn Mûsâ al-Khowârizmỵ, Mohammed, 26 Identity(ies), 82 multiplicative, 11 Identity function, 237 Identity matrix, 588–89 Identity Properties, 589 of real numbers, 10–11 Image (value) of function, 201, 203–6 Imaginary part of complex number, 105 Imaginary unit (i), 104–5 Implicit form of function, 206 Improper rational expression, 598 Improper rational function, 347 Inconsistent systems of equations, 542, 543, 548, 550 containing three variables, 550 containing two variables, 546–47 Cramer’s Rule with, 577 matrices to solve, 565 Increasing functions, 224–25, 228 Increasing linear functions, 275 Independent systems of equations, 543 Independent variable, 204 Index/indices of radical, 73, 78 row and column, 556, 581 of sum, 641 Induction, mathematical, 664–67 Extended Principle of, 667 principle of, 664–65, 667 proving statements using, 664–66 Inequality(ies), 119–34 absolute value, 130–32 combined, 124–26 equivalent, 121, 123, 126 graphing, 18–19, 614–18 on graphing utility, A9 linear inequalities, 615–16 steps for, 615 interval notation for, 120–21 involving quadratic functions, 309–11 nonstrict, 18 in one variable, 123 polynomial, 368–69 algebraically and graphically solving, 368–69 steps for solving, 369 properties of, 121–23 rational, 369–71 steps for solving, 370–71 satisfying, 614 sides of, 18 solutions of, 123 solving, 123–26 strict, 18 systems of, 614–21 graphing, 615–18 in two variables, 614 I9 I10 Index Inequality symbols, 18 Infinite geometric series, 656–57 Infinite limit, 331 Infinite sets, 680 Infinity, limits at, 331 Inflation, 474 Inflection point, 481 Initial value of exponential function, 423 Input to relation, 200 Integers, 4, dividing, 45 factoring over the, 50 Intercept(s) of circle, 184 from an equation, 160 from a graph, 159 graphing an equation in general form using, 174–75 graphing utility to find, A5–A6 from graph of linear equation, 163 graph of lines using, 174–75 Intercepts, of quadratic function, 292–95 Interest compound, 466–72 computing, 466–68 continuous, 469 defined, 466 doubling or tripling time for money, 471–72 effective rates of return, 469–70 formula, 467–68 future value of lump sum of money, 466–69 present value of lump sum of money, 470–71 problems involving, 136–37 rate of, 136, 466 effective, 469–70 simple, 136, 466 Intermediate Value Theorem, 382–84 Internal Revenue Service Restructuring and Reform Act (RRA), 233 Intersection of sets, 2–3 Interval notation, 120–21 Intervals confidence, 133 writing, using inequality notation, 121 Inverse additive, 11, 65 of matrix, 589–92 finding, 589–92 multiplying matrix by, 589–91 solving system of linear equations using, 593 multiplicative, 11 Inverse functions, 411–17 defined by a map or an ordered pair, 411–13 domain of, 412 of domain-restricted function, 416–17 finding, 411–13 defined by an equation, 414–17 graph of, 413–14 range of, 412 verifying, 413 Inverse variation, 189–90, 191 Irrational numbers, 4, 5, 15, 104 decimal representation of, Irreducible quadratic factor, 381, 602–3 Isosceles triangle, 156 Joint variation, 190–91 Jordan, Camille, 540 Kepler, Johannes, 191 Kepler’s Third Law of Planetary Motion, 196 Khayyám, Omar, 673 Kirchhoff’s Rules, 554–55, 570 Kôwa, Takakazu Seki, 540 Latitude, 319 Latus rectum, 506, 507 Law of Decay, 478–79 Laws of Exponents, 21–23, 422, 431 Leading coefficient, 40, 388 Least common multiple (LCM) to add rational expressions, 66–68 to add two quotients, 14 Left endpoint of interval, 120 Left stochastic transition matrix, 597 Legs of triangle, 30 Leibniz, Gottfried Wilhelm, 199, 540 Lensmaker’s equation, 72 Like radicals, 74–75 Like terms, 40 Limits, 331, 344 infinite, 331 at infinity, 331 Line(s), 167–82 of best fit, 284–85 coincident, 543 equations of secant, 229 family of, 181 graphing given a point and the slope, 170 using intercepts, 174–75 horizontal, 171–72 number line, 17–18 point-slope form of, 171–72 slope of, 167–70, 173 containing two points, 168 from linear equation, 173 tangent, 187 vertical, 167 y-intercept of, 173 Index Linear algebra, 580 Linear equation(s), 82–92 applied problems involving, 87–89 on calculators, 85 defined, 175 in general form, 174–75 given two points, 172 historical feature on, 89 for horizontal line, 171–72 in one variable, 82, 84 for parallel line, 175–76 for perpendicular line, 176–77 slope from, 173 in slope-intercept form, 172–73 solving equations that lead to, 86–87 steps for solving, 88 for vertical line, 170–71 Linear factors, 598–602 nonrepeated, 598–99 repeated, 600–601 Linear functions, 272–81 average rate of change of, 272–75 building from data, 282–88 defined, 272 graphing utility to find the line of best fit, 284–85 graph of, 272 identifying, 423–25 increasing, decreasing, or constant, 275 nonlinear relations vs., 283–84 scatter diagrams, 282–83 Linear models from data, 282–88 from verbal descriptions, 276–78 Linear programming problems, 540, 621–28 maximum, 625–26 minimum, 624–25 setting up, 622 solution to, 623–24 location of, 624 solving, 622–26 in two variables, 622 Line segment, midpoint of, 153–54 Local maxima and local minima of functions, 225–26 Logarithmic equations, 459–65 defined, 444 solving, 444–45, 459–61 Logarithmic functions, 437–50 changing between logarithmic expressions and exponential expressions, 438 defined, 438 domain of, 439–40 evaluating, 438–39 fitting to data, 488–89 graph of, 440–43 base not 10 or e, 456 properties of, 440, 446 range of, 439 Logarithms, 450–58 on calculators, 455 common (log), 442, 455, 456 evaluating, with bases other than 10 or e, 455–56 historical feature on, 456 logarithmic expression as single, 453–54 logarithmic expression as sum or difference of, 453 natural (ln), 441, 455, 456 properties of, 450–56 establishing, 451 proofs of, 451–52 summary of, 456 using, with even exponents, 461 relating to exponents, 438 Logistic functions, fitting to data, 489–90 Logistic models, 481–83 defined, 481 domain and range of, 481 graph of, 481 properties of, 481 Loudness, 449 Louis, Spiridon, 149 Lowest terms, rational function in, 343, 346 Magnitude, of earthquake, 450 Mandel, Howie, 679 Mapping, 200 Marathon, 149 Marginal cost, 299 Marginal propensity to consume, 663 Markov chains, 634 Mathematical induction, 664–67 Extended Principle of, 667 principle of, 664–65, 667 proving statements using, 664–66 Mathematical modeling, 134–35 Matrix/matrices, 540, 556–70, 580–97 arranging data in, 581 augmented, 557–58 in row echelon form, A9–A10 coefficient, 557 defined, 556, 581 entries of, 556, 581, 588 equal, 582 examples of, 581 graphing utilities for, 583 historical feature on, 594 identity, 588–89 inverse of, 589–92 finding, 589–92 multiplying matrix by, 589–91 solving system of linear equations using, 593 I11 I12 Index Matrix/matrices (Continued) left stochastic transition, 597 m by n, 581 nonsingular, 589, 591 product of two, 584–89 in reduced row echelon form, 563–67 row and column indices of, 556, 581 in row echelon form, 559–67 row operations on, 558–59 scalar multiples of, 583–84 singular, 589 to solve system of linear equations, 559–67 square, 581 sum and difference of two, 582–83 transition, 634 zero, 583 Maxima of functions absolute, 226–27 local, 225–26 Maximum value of a quadratic function, 296 Mean arithmetic, 129 geometric, 129 harmonic, 129 Mean distance, 523, 539 Medians of triangle, 155 Midpoint formula, 153–54 Minima of functions absolute, 226–27 local, 225–26 Minimum value of a quadratic function, 296 Minors, 574–75 Mixed numbers, Mixture problems, 137–38 Model(s), 134–35 linear from data, 282–88 from verbal descriptions, 276–78 using direct variation, 189, 191 using inverse variation, 189–90, 191 using joint variation or combined variation, 190–91 Monomial(s), 39 common factors, 50 degree of, 39, 47 examples of, 39 recognizing, 40 in two variables, 47 Monter, 181 Motion, uniform, 138–39 Multiplication, of complex numbers, 106–7 in order of operation, of polynomials, 42 of quotients, 13–14 of rational expressions, 63–64 scalar, 583–84 by zero, 12 Multiplication principle of counting, 682–83 Multiplication properties, 18 for inequalities, 122–23 Multiplicative identity, 11 Multiplicative inverse, 11 Multiplier, 663 Mutually exclusive events, 698–99 Napier, John, 456 Nappes, 504 Natural logarithms (ln), 441, 455, 456 Natural numbers (counting numbers), 4, 5, 666 Negative numbers real, 18 square roots of, 109–10 Newton’s Law of Cooling, 479–80, 484 Newton’s Law of Heating, 485 Newton’s Law of universal gravitation, 374 Newton’s Method, 352 Niccolo of Brescia (Tartaglia), 384 Nonlinear equations, systems of, 605–13 elimination method for solving, 606–10 historical feature on, 610 substitution method for solving, 605–6 Nonlinear functions, 273 Nonlinear inequalities, systems of, 617–18 Nonlinear relations, 283–84 Nonnegative property of inequalities, 121 Nonsingular matrix, 589, 591 Nonstrict inequalities, 18 nth roots, 73–74 historical feature, 78 rationalizing the denominator, 75 simplifying, 73 simplifying radicals, 74–75 Null (empty) sets, 2, 680 Number lines, 17–18, 19–20 Numbers classification of, 4–5 Fibonacci, 641 irrational, 4, 5, 6, 15 mixed, natural (counting), 4, 5, 666 negative, 18 rational, 4, triangular, 646 whole, Numerator, 4, 62 Numerical expressions, 8–9 Objective function, 622–26 Oblique asymptote, 346, 348–50, 356 Index Odd functions determining from graph, 223 identifying from equation, 224 Olympics, first modern (1896), 149 One-to-one functions, 408–11 defined, 409 horizontal-line test for, 410–11 Open interval, 120 Opens down, 290 Opens up, 290 Optimization, quadratic functions and, 300 Orbits elliptical, 503 planetary, 523 Ordered pair(s), 150 inverse function defined by, 411–13 as relations, 200–201 Order of operations, Ordinary annuity, 659 Ordinate (y-coordinate), 150 Origin, 150 distance from point to, 257–58 of real number line, 17 symmetry with respect to, 160–62 Outcome of probability, 694 equally likely, 696–97 Output of relation, 200 Parabola, 290–92, 504, 505–13 axis of symmetry of, 290, 505 defined, 505 directrix of, 505 family of, 255 focus of, 505 graphing equation of, 506 solving applied problems involving, 510–11 with vertex at (h, k), 508–9 with vertex at the origin, 505–8 finding equation of, 507–8 focus at (a, 0), a > 0, 506–7 vertex of, 290, 505 Paraboloids of revolution, 503, 510 Parallel lines, 175–76 Parentheses, order of operations and, Partial fraction decomposition, 540, 597–604 defined, 598 where denominator has nonrepeated irreducible quadratic factor, 602–3 where denominator has only nonrepeated linear factors, 598–99 where denominator has repeated irreducible quadratic factors, 603 where denominator has repeated linear factors, 600–602 Partial fractions, 598 Participation rate, 213 Pascal, Blaise, 670, 700 Pascal triangle, 670, 673 Payment period, 466 Peano, Giuseppe, 701 Pendulum period of, 80, 192 simple, 192 Perfect cubes, 44 Perfect roots, 73 Perfect squares, 43, 51–52 Perihelion, 523, 539 Perimeter, formulas for, 31 Period, of pendulum, 80, 192 Permutations, 685–88 computing, 688 defined, 685 distinct objects without repetition, 686–88 distinct objects with repetition, 686 involving n nondistinct objects, 690–91 Phones, cellular, 199 Piecewise-defined functions, 239–41 Pitch, 181 Pixels, A1 Planetary motion, Kepler’s Third Law of, 196 Planets, orbit of, 523 Plotting points, 150 graph equations by, 157–59 Point(s) coordinate(s) of on number line, 17 on graphing utility, A2 corner, 619 distance between two, 151 distance from the origin to, 257–58 feasible, 622, 623–24 inflection, 481 plotting, 150 graph equations by, 157–59 of tangency, 187 turning, 329–30 Point-slope form of equation of line, 171–72 Polynomial(s), 39–58 adding, 41 degree of, 40, 47, 320–24 odd, 381, 389 second-degree, 52–53 dividing, 44–47, 375–77 synthetic division, 58–62 examples of, 40–41 factoring, 49–58 Ax2 + Bx + C, 54–55 difference of two squares and the sum and the difference of two cubes, 50–51 by grouping, 53–54 perfect squares, 51–52 x2 + Bx + C, 52–53 I13 I14 Index Polynomial(s) (Continued) multiplying, 42 prime, 50, 53 recognizing, 40–41 solving, 381 special products formulas, 43–44 in standard form, 40 subtracting, 41–42 terms of, 40 in two variables, 47 zero, 40 Polynomial functions, 320–42 complex, 388 complex zeros of, 388, 391 Conjugate Pairs Theorem, 389 defined, 388 finding, 391 polynomial function with specified zeros, 390 cubic models from data, 336–37 defined, 320 end behavior of, 330–32 graph of, 322–37 analyzing, 332–36 end behavior of, 330–32 smooth and continuous, 321 turning points of, 329–30 using bounds on zeros, 382 using transformations, 324 using x-intercepts, 326–27 historical feature on, 384 identifying, 320–24 multiplicity of, 325–27 behavior near zero and, 327–28 real zeros (roots) of, 325–27, 374–87 finding, 379–80 Intermediate Value Theorem, 382–84 number of, 377 Rational Zeros Theorem, 378, 391 Remainder Theorem and Factor Theorem, 375–77 repeated, 326 theorem for bounds on, 381–82 solving, 379–80 unbounded in the negative direction, 331 Polynomial inequalities, 368–69 algebraically and graphically solving, 368–69 steps for solving, 369 Population, world, 636 Positive real numbers, 18 Power(s), 22 of i, 108–9 log of, 452 Power functions, 321–24 exponential function vs., 423 graph of, 322–23 of odd degree, 323 properties of, 323 Present value, 468, 470–71 Price, equilibrium, 277–78 Prime polynomials, 50, 53 Principal, 136, 466 Principal nth root of real number, 73 Principal square root, 23, 109 Probability(ies), 634, 694–704 Complement Rule to find, 699–700 compound, 697 constructing models, 694–96 defined, 694 of equally likely outcomes, 696–97 of event, 696 mutually exclusive, 698–99 historical feature on, 700–701 outcome of, 694 sample space, 694 of union of two events, 698–99 Product(s), of complex numbers, 106 log of, 452 special, 43–44, 47 of two functions, 208 of two matrices, 584–89 Product function, 208 Prolate spheroid, 523 Proper rational expressions, 598 Proper rational function, 347–48 Proper subsets, 680 Proportionality, constant of, 189, 190 Pure imaginary number, 105 Pythagorean Brotherhood, 15 Pythagorean Theorem, 30–31 applying, 31 converse of, 30–31 proof of, 35 Pythagorean triples, 38 Quadrants, 150 Quadratic equations, 92–104 applied problems involving, 99–100 completing the square to solve, 95–96 in complex number system, 109–11 defined, 93 discriminant of, 97 negative, 109 factoring, 93–95, 116–17 historical feature on, 100 procedure for solving, 99 quadratic formula for, 96–99, 110 Square Root Method for solving, 94 in standard form, 93 Quadratic factors, irreducible, 381, 602–3 Index Quadratic formula, 96–99, 110 Quadratic functions, 288–99 defined, 289 graph of properties of, 292–96 steps for, 296 using its vertex, axis, and intercepts, 292–96 using transformations, 290–92 inequalities involving, 309–11 maximum or minimum value of, 296, 300 optimizations and, 300 vertex and axis of symmetry of, 292–96 Quadratic models, 300–309 from data, 304–5 from verbal descriptions, 300–304 Quantity, equilibrium, 277–78 Quantity demanded, 277–78 Quantity supplied, 277–78 Quotient(s), 7, 12, 45, 375 arithmetic of, 13–14 of complex numbers in standard form, 107 difference, 205, 234, 436 log of, 452 subtraction of, 13–14 synthetic division to find, 60–61 of two functions, 209 Radical equations, 113–19 defined, 113 graphing utility to solve, A7 solving, 113–14 Radicals, 73 fractional exponents as, 76 index of, 73, 78 like, 74–75 properties of, 74 rational exponents defined using, 76 simplifying, 74–75 Radical sign, 23, 78 Radicand, 73 Radioactive decay, 478–79 Radius, 182 Range, 201 of absolute value function, 238 of constant function, 237 of cube function, 237 of cube root function, 237 of greatest integer function, 238 of identity function, 237 of inverse function, 412 of logarithmic function, 439 of logistic models, 481 of one-to-one function, 409 of reciprocal function, 238 of square function, 237 of square root function, 237 Rate of change, average, 168, 228–30, 272–75 of linear and exponential functions, 424–25 Rate of interest, 136, 466 Rates of return, effective, 469–70 Ratio common, 653 golden, 645–46 conjugate, 646 Rational exponents, 76–77 Rational expressions, 62–72 adding and subtracting, 64–66 least common multiple (LCM) method for, 66–68 application of, 70 complex, 68–70 defined, 62 improper, 598 multiplying and dividing, 63–64 proper, 598 reducing to lowest terms, 62–63 Rational functions, 342–67 applied problems involving, 364 asymptotes of, 345–46 horizontal, 345–46, 347–50 vertical, 346–47 defined, 342, 343 domain of, 343–46 graph of, 353–67 analyzing, 353–63 constructing rational function from, 363–64 end behavior of, 346 using transformations, 344 with a hole, 361–63 improper, 347 in lowest terms, 343, 346 proper, 347–48 unbounded in positive direction, 344 Rational inequalities, 369–71 steps for solving, 370–71 Rationalizing the denominator, 75 Rational numbers, 4, 5, 104, 342 Rational Zeros Theorem, 378, 391 Real number(s), 2–17, 104 approximating decimals, conjugate of, 108 defined, historical feature on, 15 number line representation of, 17–18 numerical expressions, 8–9 positive and negative, 18 principal nth root of, 73 properties of, 9–14 square of, 104 I15 I16 Index Real number line, 17–18 distance on, 19–20 Real part of complex number, 105 Real zeros (roots) of polynomial functions, 374–87 finding, 379–80 Intermediate Value Theorem, 382–84 number of, 377 Rational Zeros Theorem, 378, 391 Remainder Theorem and Factor Theorem, 375–77 repeated, 326 theorem for bounds on, 381–82 Reciprocal, 11 of complex number in standard form, 107 Reciprocal function, 238 Reciprocal property for inequalities, 123, 126 Rectangle, area and perimeter of, 31–32 Rectangular (Cartesian) coordinate system, 149–51 Recursive formula, 640–41 for arithmetic sequences, 648–49 terms of sequences defined by, 640–41 Reduced row echelon form, 563–67 Reflections about x-axis or y-axis, 250–51 Reflexive property, Relation(s), 200 defined, 200 as function, 200–203 input to, 200 nonlinear, 283–84 ordered pairs as, 200–201 Relative maxima and minima of functions, 225–26 Remainder, 45, 375 synthetic division to find, 60–61 Remainder Theorem, 375–77 Repeated zeros (solutions), 93, 326 Review, 1–80 of algebra, 17–29 distance on the real number line, 19–20 domain of variable, 21 evaluating algebraic expressions, 20 graphing calculator to evaluate exponents, 24 graphing inequalities, 18–19 historical feature, 26 Laws of Exponents, 21–23 multiplication properties of positive and negative numbers, 18 real number line, 17–18 scientific notation, 24–26 square roots, 23–24 of geometry, 30–38 formulas, 31–32 Pythagorean theorem and its converse, 30–31, 35 of nth roots, 73–74 historical feature, 78 rationalizing the denominator, 75 simplifying, 73 simplifying radicals, 74–75 of polynomials, 39–58 adding, 41 dividing, 44–47 factoring, 49–58 monomials, 39 multiplying, 42 recognizing, 40–41 special products formulas, 43–44 subtracting, 41–42 synthetic division of, 58–62 in two variables, 47 of rational exponents, 76–77 of rational expressions, 62–72 adding and subtracting, 64–66 application of, 70 complex, 68–70 multiplying and dividing, 63–64 reducing to lowest terms, 62–63 of real numbers, 2–17 approximating decimals, defined, historical feature on, 15 number line representation of, 17–18 numerical expressions, 8–9 properties of, 9–14 Rhind papyrus, 660 Richter scale, 450 Right angle, 30 Right circular cone, 504 Right circular cylinder, volume and surface area of, 32 Right endpoint of interval, 120 Right triangles, 30 Rise, 167 Root(s), 82 perfect, 73 Root of multiplicity (double root), 93 Roster method, Rounding, Row echelon form, 559–67 augmented matrix in, A9–A10 reduced, 563–67 Row index, 556, 581 Row operations, 558–59 Row vector, 585 Rudolff, Christoff, 78 Ruffini, P., 384 Rules of Signs, 12 Run, 167 Rutherford, Ernest, 536 Sample space, 694 Satisfying equations, 82, 157 Satisfying inequalities, 614 Index Scalar, 583 Scalar multiples of matrix, 583–84 Scale of number line, 17 Scatter diagrams, 282–83 Schroeder, E., 701 Scientific calculators, Scientific notation, 24–26 Secant line, 229–30 Second-degree polynomials, 52–53 Sequences, 637–64 annuity problems, 659–60 arithmetic, 647–52 common difference in, 647 defined, 647 determining, 647–48 formula for, 648–49 nth term of, 648 recursive formula for, 648–49 sum of, 649–51 defined, 637 factorial symbol, 639–40 Fibonacci, 641, 645 geometric, 653–56 common ratio of, 653 defined, 653 determining, 653–54 formula for, 654–55 nth term of, 654–55 sum of, 655–56 graph of, 637–38 historical feature on, 660 from a pattern, 639 properties of, 642 summation notation, 641–42 sum of, 642–43 terms of, 637–39 alternating, 639 defined by a recursive formula, 640–41 general, 638 Set(s), 2–3, 15 complement of, correspondence between two, 200 defined, 680 disjoint, elements of, 2, 680–82 empty (null), 2, 680 equal, 2, 680 finite, 680 infinite, 680 intersection of, 2–3 of numbers, 4–5 subsets of, 680 proper, 680 union of, 2–3 universal, 3, 681 Set-builder notation, Set theory, 15 Shannon’s diversity index, 448 Shifts, graphing functions using vertical and horizontal, 244–47, 251 Side-angle-side case of congruent triangle, 33 Side-angle-side case of similar triangle, 34 Sides of equation, 82, 157 of inequality, 18 Side-side-side case of congruent triangle, 33 Side-side-side case of similar triangle, 34 Signs, Rules of, 12 Similar triangles, 33–35 Simple interest, 136, 466 Simple pendulum, 192 Simplifying complex rational expressions, 68–70 expressions with rational exponents, 76–77 nth roots, 73 radicals, 74–75 Simpson’s rule, 307 Sine function, hyperbolic, 437 Singular matrix, 589 Slope, 167–70, 173 containing two points, 168 graphing lines given, 170 from linear equation, 173 of secant line, 229 Slope-intercept form of equation of line, 172–73 Smooth graph, 321 Solution(s), 82 extraneous, 113 of inequalities, 123 of linear programming problems, 623–24 location of, 624 repeated, 93, 326 of systems of equations, 542, 548–49 Solution set of equation, 82 Special products, 43–44, 47 Sphere, volume and surface area of, 32 Spheroid, prolate, 523 Square(s) of binomials (perfect squares), 43, 50 completing the, 56 difference of two, 43, 50–51 perfect, 43, 51–52 Square function, 237 Square matrix, 581 Square root(s), 23–24, 73 of negative numbers, 109–10 principal, 23, 109 Square root function, 234–35, 237 Square Root Method, 94 Standard deviation, 129 I17 I18 Index Standard form complex numbers in, 105, 107–9 of equation of circle, 182–83 polynomials in, 40 quadratic equations on, 93 Statements, writing using symbols, Step function, 239 Stevin, Simon, 15 Stirling’s formula, 674 Stock valuation, 271 Stretches, graphing functions using, 247–49, 251 Strict inequalities, 18 Subscripted letters, 637 Subsets, 2, 680 proper, 680 Substitution, principle of, Substitution method, 540, 543–44 systems of nonlinear equations solved using, 605–6 Subtraction, of complex numbers, 105 in order of operations, of polynomials, 41 of quotients, 13–14 of rational expressions, 64–66 least common multiple (LCM) method for, 66–68 Sum, of arithmetic sequences, 649–51 of complex numbers, 105 of geometric sequences, 655–56 index of, 641 of infinite geometric series, 657 of logarithms, 453 of sequences, 642–43 of two cubes, 44, 50 of two functions, 208 of two matrices, 582–83 Sum function, 208 Summation notation, 641–42 Surface area, formulas for, 32 Sylvester, James J., 594 Symbols, writing statements using, Symmetric property, Symmetry, 160–62 axis of of parabola, 290 of quadratic function, 292–96 axis of, of parabola, 505 graphing utility to check for, A5–A6 with respect to origin, 160–62 with respect to the x-axis, 160–61, 162 with respect to the y-axis, 161, 162 Synthetic division, 58–62 Systems of equations consistent, 542, 548 dependent, 543 containing three variables, 550–51 containing two variables, 547–48 Cramer’s Rule with, 577 equivalent, 545 graphing, 543 inconsistent, 542, 548, 550 containing three variables, 550 containing two variables, 546–47 Cramer’s Rule with, 577 independent, 543 solutions of, 542, 548–49 Systems of inequalities, 614–21 graphing, 615–18 bounded and unbounded graphs, 618 vertices or corner points, 619 Systems of linear equations, 541–80 consistent, 543, 548 defined, 542–43 dependent, 543 containing three variables, 550–51 containing two variables, 547–48 matrices to solve, 563–65 determinants, 571–80 cofactors, 575 Cramer’s Rule to solve a system of three equations containing three variables, 576–77 Cramer’s Rule to solve a system of two equations containing two variables, 572–74 minors of, 574–75 properties of, 577–78 by 3, 574–76 by 2, 571, 577–78 elimination method of solving, 545–46, 547–48 equivalent, 545 examples of, 541–42 graphing, 543 inconsistent, 543, 548, 550 containing three variables, 550 containing two variables, 546–47 matrices to solve, 565 independent, 543 partial fraction decomposition, 597–604 defined, 598 where denominator has a nonrepeated irreducible quadratic factor, 602–3 where denominator has only nonrepeated linear factors, 598–99 where denominator has repeated irreducible quadratic factors, 603 where denominator has repeated linear factors, 600–602 Index solution of, 542, 548–49 solving, 542 with graphing utility, A9–A10 substitution method of, 543–44 three equations containing three variables, 548–49 Systems of nonlinear equations, 605–13 elimination method for solving, 606–10 historical feature on, 610 substitution method for solving, 605–6 Systems of nonlinear inequalities, graphing, 617–18 Tables, on graphing utility, A4 Tangency, point of, 187 Tangent line, 187 Greek method for finding, 187 Tartaglia (Niccolo of Brescia), 384 Terms like, 40 of polynomial, 40 of sequences, 637–39 alternating, 639 defined by a recursive formula, 640–41 general, 638 by determinants, 574–76 TI-84, A8n TI-84 Plus, A3, A9, A10 Transformations, 508, 519, 531 combining, 247, 252 compressions and stretches, 247–49, 251 defined, 244 graphs using of exponential functions, 428, 429–30 of polynomial functions, 324 of quadratic functions, 290–92 of rational functions, 344 reflections about the x-axis or y-axis, 250–51 vertical and horizontal shifts, 244–47, 251 Transition matrix, 634 Transitive property, Transverse axis, 524 Tree diagram, 683 Triangle(s) area of, 31 congruent, 35 equilateral, 155 error, 156 isosceles, 156 legs of, 30 medians of, 155 Pascal, 670, 673 right, 30 similar, 33–35 Triangular addition, 672 Triangular number, 646 Trinomials, 40 factoring, 52–53, 54–55 Truncation, Turning points, 329–30 by determinants, 571 proof for, 577–78 Unbounded graphs, 618 Unbounded in positive direction, 344 Unbounded in the negative direction, polynomial functions, 331 Uniform motion, 138–39 Uninhibited growth, 476–78 Union of sets, 2–3 of two events, probabilities of, 698–99 Unit circle, 183 Universal sets, 3, 681 Value (image) of function, 201, 203–6 Variable(s), 20, 39 complex, 388 dependent, 204 domain of, 21 independent, 204 Variable costs, 180 Variation, 189 combined, 190–91 direct, 189, 191 inverse, 189–90, 191 joint, 190–91 Vector(s) column, 585 row, 585 Venn diagrams, Verbal descriptions linear models from, 276–78 quadratic models from, 300–304 Vertex/vertices, 619 of cone, 504 of ellipse, 514 of hyperbola, 524 of parabola, 290, 505 of quadratic function, 292–96 Vertical asymptote, 345, 346–47 Vertical line, 167 Vertical-line test, 214–15 Vertically compressed or stretched graphs, 24749 Vertical shifts, 24447, 251 Viốte, Franỗois, 100 Viewing rectangle, 151, A1–A3 setting, A1 Vinculum, 78 Volume, formulas for, 32 I19 ... 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